Quantum computing: From linear algebra to physical by Mikio Nakahara, Tetsuo Ohmi

By Mikio Nakahara, Tetsuo Ohmi

Protecting either thought and revolutionary experiments, Quantum Computing: From Linear Algebra to actual Realizations explains how and why superposition and entanglement give you the huge, immense computational energy in quantum computing. This self-contained, classroom-tested booklet is split into sections, with the 1st dedicated to the theoretical elements of quantum computing and the second one fascinated with a number of applicants of a operating quantum desktop, comparing them in keeping with the DiVincenzo standards.

Topics partially I

  • Linear algebra
  • Principles of quantum mechanics
  • Qubit and the 1st program of quantum details processing—quantum key distribution
  • Quantum gates
  • Simple but elucidating examples of quantum algorithms
  • Quantum circuits that enforce crucial transforms
  • Practical quantum algorithms, together with Grover’s database seek set of rules and Shor’s factorization set of rules
  • The nerve-racking factor of decoherence
  • Important examples of quantum error-correcting codes (QECC)

Topics partially II

  • DiVincenzo standards, that are the factors a actual procedure needs to fulfill to be a candidate as a operating quantum computer
  • Liquid nation NMR, one of many well-understood actual systems
  • Ionic and atomic qubits
  • Several sorts of Josephson junction qubits
  • The quantum dots recognition of qubits

Looking on the ways that quantum computing can turn into fact, this ebook delves into adequate theoretical historical past and experimental study to help an intensive realizing of this promising field.

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46) 24 QUANTUM COMPUTING where U ∈ U(m), V ∈ U(n) and Σ is an m × n matrix whose diagonals are nonnegative real numbers, called the singular values, while all the off diagonal components are zero. The matrix Σ is called the singular value matrix. 46) is called the singular value decomosition and is often abbreviated as SVD. We now sketch the proof of the decomposition. Let us assume m > n for definiteness. Consider the eigenvalue problem of an n × n Hermitian matrix A† A; A† A|λi = λi |λi (1 ≤ i ≤ n), where λi is a nonnegative real number, where nonnegativity follows from the observation λi = λi |λi |λi = λi |A† A|λi = A|λi 2 ≥ 0.

The above equation being a linear equation, the norm of the eigenvector cannot be fixed. Of course, it is always possible to normalize |v such that |v = 1. We often use the symbol |λ for an eigenvector corresponding to an eigenvalue λ to save symbols. Let {|ek } be an orthonormal basis in Cn and let ei |A|ej = Aij and vi = ei |v be the components of A and |v with respect to the basis. Then the component expression for the above equation is obtained from |ei ei |A|ej ej |v = A|v = i,j Aij vj |ei i,j as Aij vj = λvi .

9) The matrix exponential function in this equation is evaluated with the help of Eq. 44) and we find |ψ(t) = cos ωt/2 i sin ωt/2 i sin ωt/2 cos ωt/2 cos ωt/2 1 0 = . 10) i sin ωt/2 Suppose we measure the observable σz . Note that |ψ(t) is expanded in terms of the eigenvectors of σz as |ψ(t) = cos ω ω t|σz = +1 + i sin t|σz = −1 . 2 2 Therefore we find the spin is in the spin-up state with the probability P↑ (t) = cos2 (ωt/2) and in the spin-down state with the probability P↓ (t) = sin2 (ωt/2) as depicted in Fig.

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